We consider
the continuous Laplacian on infinite locally finite networks under
natural transition conditions as continuity at the ramification
nodes and Kirchhoff flow conditions at all vertices. It is well
known that one cannot reconstruct the shape of a finite network by
means of the eigenvalues of the Laplacian on it. The same is shown
to hold for infinite graphs in a $L^\infty$-setting. Moreover,
the occurrence of eigenvalue multiplicities with eigenspaces
containing subspaces isomorphic to $\l^\infty(\ZZ)$ is
investigated, in particular in trees and periodic graphs.